High School

We appreciate your visit to The sum of the first 7 terms of an arithmetic progression AP is 182 If its 4th and 17th terms are in the ratio 1. This page offers clear insights and highlights the essential aspects of the topic. Our goal is to provide a helpful and engaging learning experience. Explore the content and find the answers you need!

The sum of the first 7 terms of an arithmetic progression (AP) is 182. If its 4th and 17th terms are in the ratio 1:5, find the AP.

Answer :

The first term a of the A.P. is 1, and the common difference d is 4.

Therefore, the A.P. is 1, 5, 9, 13, 17, 21, 25, ...

To find the first term and the common difference of the arithmetic progression (A.P.), let's denote the first term as a and the common difference as d .

The sum of the first seven terms of an A.P. is given by the formula:

[tex]\[ S_n = \frac{n}{2}[2a + (n - 1)d] \][/tex]

where [tex]\( S_n \)[/tex] is the sum of the first n terms.

Given that the sum of the first seven terms is 182, we can write:

[tex]\[ 182 = \frac{7}{2}[2a + (7 - 1)d] \]\[ 182 = 7[2a + 6d] \]\[ 26 = 2a + 6d \]\[ 13 = a + 3d \][/tex]

Given that the 4th and 17th terms are in the ratio 1:5, we can express this as:

[tex]\[ \frac{a + 3d}{a + 16d} = \frac{1}{5} \][/tex]

Cross-multiplying:

[tex]\[ 5(a + 3d) = a + 16d \]\[ 5a + 15d = a + 16d \]\[ 4a = d \][/tex]

Now, we have two equations (equations 1 and 2) which can be solved simultaneously to find the values of a and d .

From equation (2), we can express d in terms of a :

[tex]\[ d = \frac{4a}{1} \]\[ d = 4a \][/tex]

Substituting this value of d into equation (1), we get:

[tex]\[ 13 = a + 3(4a) \]\[ 13 = a + 12a \]\[ 13 = 13a \]\[ a = 1 \][/tex]

Now, substituting a = 1 into equation (2), we get:

[tex]\[ d = 4(1) \]\[ d = 4 \][/tex]

So, the first term a of the A.P. is 1, and the common difference d is 4.

Therefore, the A.P. is 1, 5, 9, 13, 17, 21, 25, ...

Thanks for taking the time to read The sum of the first 7 terms of an arithmetic progression AP is 182 If its 4th and 17th terms are in the ratio 1. We hope the insights shared have been valuable and enhanced your understanding of the topic. Don�t hesitate to browse our website for more informative and engaging content!

Rewritten by : Barada